# Linear Programming Dual

May 5th, 2015
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Every linear programming problem has associated with it another linear programming problem and that the two problems have such a close relationship that whenever one problem is solved, the other is solved as well. The original LPP is called the primal problem and the associated LPP is called the dual problem. Together they are called a \dual pair" (primal + dual) in the sense that the dual of the dual will again be the primal.

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Linear Programming DualChapter 5DUALITY5.1 The Dual ProblemsEvery linear programming problem has associated with it another linear programming problem and that the two problems have such a close relationship that whenever one problem is solved, the other is solved as well. The original LPP is called the primal problem and the associated LPP is called the dual problem. Together they are called a \dual pair" (primal + dual) in the sense that the dual of the dual will again be the primal.Example 5.1. (The Diet Problem) How can a dietician design the most economical diet that satises the basic daily nutritional requirements for a good health? For simplicity, we assume that there are only two foods F1 and F2 and the daily nutrition required are N1, N2 and N3. The unit cost of the foods and their nutrition values together with the daily requirement of each nutrition are given in the following table.F1 F2 Daily RequirementCost 120 180 {N1 1 1 10N2 2 4 24N3 3 6 32Let xj , j = 1; 2 be the number of units of Fj that one should eat in order to minimize the cost and yet fulll the daily nutrition requirement. Thus the problem is to select the xj such that min x0 = 120x1 + 180x2subject to the nutritional constraints:8><>:x1 + x2 102x1 + 4x2 243x1 + 6x2 32and the non-negativity constraints: xj 0, j = 1; 2. In matrix form, we have min x0 = cT xsubject to(Ax bx 012 Chapter 5. DUALITYwherec =120180; x =x1x2; b =241024

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